Mathematics as an optional subject has one of the highest scoring potential in UPSC CSE, provided you have done your preparation adequately and are able to write your answers accurately. The subject is less unpredictable compared to the humanities subjects because of its objective and almost consistent method of research and problem-solving. This makes it a good alternative for engineers, aspirants with a background in science, or Mathematics graduates who already have a technical mindset.

Reading and thoroughly understanding the theories and concepts is the **ideal approach to start preparing for UPSC Maths optional subject**. The three main pillars of subject preparation are reading, revision, and assessments. While the exam occasionally includes questions about theorem proofs, particularly in Paper II, the ability to develop a theory or theorem from concept to proof is essential.

Furthermore, there is a strong probability that questions from the previous year will be asked again. The fact that Mathematics is utterly disconnected from current events taking place across the world is another advantage of choosing it as an optional subject. Even yet, it shields aspirants from the tedious character of General Studies and presents them favourably in the Interview stage.

Mathematics is also known as ‘Queen of Sciences’, it therefore is highly objective in nature and because it has a static syllabus, it makes it a lucrative optional subject. If one has a sound basic knowledge of topics like Differentiation, Integration, Matrices, etc. and has a passion for the subject, then it is a question of putting in effort in the correct direction once and then practice will help sail through the exam smoothly. However, keep in mind that not only accuracy in terms of getting correct answers matters but also the style of presentation. Hence, ** the focus must be on core fundamentals as well as the manner in which the answers are structured**. For example, quoting the name of the procedure/method being employed (Charpit’s method – Partial Differential Equations, Exact equations – Ordinary Differential Equations, Vogel’s Approximation method etc.) leads to the quality of presentation increasing manifold.

Also, learning how to write the solution in steps leads to more structured answers, which automatically leads to an increase in marks allotted to the solution, even if there is a computational mistake (step marking). This is also beneficial as the UPSC Maths Optional syllabus comprises not only applied topics like Calculus and numerical analysis but also theoretical ones like Real Analysis, Abstract Algebra etc. A grasp over theorems in various domains gives an edge over others, too, as the proof can be then written verbatim, and there is no scope for any marks being deducted as they are outside the zone of calculation mistakes. The icing on the cake is the fact that in the last few years, mathematics has done extremely well in UPSC, with toppers also attributing their high scores to their performance, specifically in optional papers.

The best time to start preparing for Mathematics as an optional subject for UPSC is at least a year before Mains in the subsequent year. At the onset, one needs to go through the entire syllabus in a detailed manner, focusing on all the topics. This is the stage at which students must focus on building strong fundamentals and get a good grasp of all the concepts. More than practice, the focus should be on clarity in terms of procedures, recognising the majority type of questions rather than aberrations and identifying individual areas of strength and weaknesses. This is also the time one should prepare a small booklet for each topic incorporating formulas used, the procedure of the methods and a list of important theorems/ lemmas being used. One may also do some PYQs at this stage.

Before Prelims, one should then do a second iteration of the syllabus in terms of revision, this time incorporating more PYQs and more theorems from topics like Algebra and Analysis. In spite of more focus on Prelims towards the end, one should not completely abandon practising Maths. Instead, one may choose topics according to their strengths and ease of understanding and continue working till the Prelims.

Immediately after the Prelims, a quick revision should follow solving PYQs or Tests based on the UPSC exam pattern to empower oneself to handle multiple modules simultaneously and in a time-bound manner. What is crucial is also the space constraint one has to manage as it can prove detrimental in case enough thought has not been given before attempting the question. This is also the time to focus on special questions, i.e. the ones which do not conform to the norms and on theorems which may have some memorising steps.

*In addition to the above (preparation strategy):*

The best strategy is to first focus on the type of questions which have to be an integral part of the UPSC Maths Optional paper. Identify which topics are asked more frequently than others. These topics should literally be on tips, so it saves time in the exam to be able to devote more time to questions which require more thinking.

Some of these topics and modules are Basis, dimension, spanning etc. in Linear algebra, Canonical forms, method of separation of variables in Partial Differential Equations, Gauss elimination method in Numerical Analysis, Simplex method, Transportation problem in Linear Programming, etc. One should then focus on theorems in topics like Linear Algebra, Real Analysis and Abstract Algebra. Lastly, one should focus on special questions from each module and, depending on their level of difficulty, revise them more often.

Due to the vast nature of the subject, one has to focus specifically on each module separately while preparing the syllabus. However, in the exam, 3-4 topics are covered in each section. Moreover, over the past several years, questions have come with no external choice in the topics as in one question has subparts, each belonging to a different module. Hence, it is imperative to practice writing sectional as well as full course tests to maximise the efficiency with which paper has to be attempted.

Not only that, but writing tests also helps to tune our minds to complete the paper in a time-bound fashion. It is easy to get stuck on one question for more time than it will be possible in the exam and get carried away. That luxury is not available in the exam, and writing tests in a fixed time limit helps in understanding how and where the speed can be increased.

Lastly, the space constraint can be more detrimental to UPSC Maths optional than most of the other optional subjects. This means one has to learn how to judiciously use the given space as well as utilise the rough space given to maximise output.

A test series program specially curated to focus on topic-wise preparation, and section-wise preparation as well as full paper preparation along with fixed time and precise space constraints is hence monumental to performing well in Maths optional and plays a pivotal role in your path to success.

**(1) Linear Algebra:**

Vector spaces over R and C, linear dependence and independence, subspaces, bases, dimensions, Linear transformations, rank and nullity, matrix of a linear transformation. Algebra of Matrices; Row and columnreduction, Echelon form, congruence and similarity; Rankof a matrix; Inverse of a matrix; Solution of system of linear equations; Eigenvalues and eigenvectors, characteristic polynomial, Cayley-Hamilton theorem, Symmetric, skew- symmetric, Hermitian, skew-Hermitian, orthogonal and unitary matrices and their eigenvalues.

**(2) Calculus:**

Real numbers, functions of a real variable, limits, continuity, differentiability, mean-value theorem, Taylor’s theorem with remainders, indeterminate forms, maxima and minima, asymptotes; Curve tracing; Functions of two or three variables; Limits, continuity, partial derivatives, maxima and minima, Lagrange’s method of multipliers, Jacobian. Riemann’s definition of definite integrals; Indefinite integrals; Infinite and improper integral; Double and triple integrals (evaluation techniques only); Areas, surface and volumes.

**(3) Analytic Geometry:**

Cartesian and polar coordinates in three dimensions, second degree equations in three variables, reduction to Canonical forms; straight lines, shortest distance between two skew lines, Plane, sphere, cone, cylinder, paraboloid, ellipsoid, hyperboloid of one and two sheets and their properties.

**(4) Ordinary Differential Equations:**

Formulation of differential equations; Equations of first order and first degree, integrating factor; Orthogonal trajectory; Equations of first order but not of first degree, Clairaut’s equation, singular solution.

Second and higher order linear equations with constant coefficients, complementary function, particular integral and general solution.

Section order linear equations with variable coefficients, Euler-Cauchy equation; Determination of complete solution when one solution is known using method of variation of parameters.

Laplace and Inverse Laplace transforms and their properties, Laplace transforms of elementary functions. Application to initial value problems for 2nd order linear equations with constant coefficients.

**(5) Dynamics and Statics:**

Rectilinear motion, simple harmonic motion, motion in a plane, projectiles; Constrained motion; Work and energy, conservation of energy; Kepler’s laws, orbits under central forces. Equilibrium of a system of particles; Work and potential energy, friction, Common catenary; Principle of virtual work; Stability of equilibrium, equilibrium of forces in three dimensions.

**(6) Vector Analysis:**

Scalar and vector fields, differentiation of vector field of a scalar variable; Gradient, divergence and curl in cartesian and cylindrical coordinates; Higher order derivatives; Vector identities and vector equation.

Application to geometry : Curves in space, curvature and torsion; Serret-Furenet's formulae.

Gauss and Stokes’ theorems, Green's indentities.

**(1) Algebra:**

Groups, subgroups, cyclic groups, cosets, Lagrange’s Theorem, normal subgroups, quotient groups, homomorphism of groups, basic isomorphism theorems, permutation groups, Cayley’s theorem.

Rings, subrings and ideals, homomorphisms of rings; Integral domains, principal ideal domains, Euclidean domains and unique factorization domains; Fields, quotient fields.

**(2) Real Analysis:**

Real number system as an ordered field with least upper bound property; Sequences, limit of a sequence, Cauchy sequence, completeness of real line; Series and its convergence, absolute and conditional convergence of series of real and complex terms, rearrangement of series.

Continuity and uniform continuity of functions, properties of continuous functions on compact sets.

Riemann integral, improper integrals; Fundamental theorems of integral calculus.

Uniform convergence, continuity, differentiability and integrability for sequences and series of functions; Partial derivatives of functions of several (two or three) variables, maxima and minima.

**(3) Complex Analysis:**

Analytic function, Cauchy-Riemann equations, Cauchy's theorem, Cauchy's integral formula, power series, representation of an analytic function, Taylor’s series; Singularities; Laurent’s series; Cauchy’s residue theorem; Contour integration.

**(4) Linear Programming:**

Linear programming problems, basic solution, basic feasible solution and optimal solution; Graphical method and simplex method of solutions; Duality.

Transportation and assignment problems.

**(5) Partial Differential Equations:**

Family of surfaces in three dimensions and formulation of partial differential equations; Solution of quasilinear partial differential equations of the first order, Cauchy’s method of characteristics; Linear partial differential equations of the second order with constant coefficients, canonical form; Equation of a vibrating string, heat equation, Laplace equation and their solutions.

**(6) Numerical Analysis and Computer Programming:**

Numerical methods: Solution of algebraic and transcendental equations of one variable by bisection, Regula-Falsi and Newton-Raphson methods, solution of system of linear equations by Gaussian Elimination and Gauss-Jorden (direct), Gauss-Seidel (iterative) methods. Newton’s (forward and backward) and interpolation, Lagrange’s interpolation.

Numerical integration: Trapezoidal rule, Simpson’s rule, Gaussian quadrature formula.

Numerical solution of ordinary differential equations : Eular and Runga Kutta methods.

Computer Programming : Binary system; Arithmetic and logical operations on numbers; Octal and Hexadecimal Systems; Conversion to and from decimal Systems; Algebra of binary numbers.

Elements of computer systems and concept of memory; Basic logic gates and truth tables, Boolean algebra, normal forms.

Representation of unsigned integers, signed integers and reals, double precision reals and long integers.

Algorithms and flow charts for solving numerical analysis problems.

**(7) Mechanics and Fluid Dynamics:**

Generalised coordinates; D’Alembert’s principle and Lagrange’s equations; Hamilton equations; Moment of inertia; Motion of rigid bodies in two dimensions.

Equation of continuity; Euler’s equation of motion for inviscid flow; Stream-lines, path of a particle; Potential flow; Two-dimensional and axisymmetric motion; Sources and sinks, vortex motion; Navier-Stokes equation for a viscous fluid.

The fact is that Mathematics optional has no overlap with the GS syllabus, and this fact can be used to one’s advantage if the choice of optional was made correctly with passion for the subject as one of the primary reasons. The study time can be clearly demarcated into GS and Maths, where primarily GS will involve learning and memorising facts (rote learning) whereas Mathematics will focus on imbibing methods and practising.

Also, since the Maths syllabus is well balanced and incorporates both theory and numericals it allows permeating into concept learning and practice similar questions. For someone who loves mathematics and has a passion for the subject will appreciate the break in the monotony that maths can offer from the never-ending process of remembering facts that is an integral part of GS preparation.

The **two basic pillars of evaluating UPSC Mathematics answers are Accuracy and presentation**. There are two kinds of questions that are asked in the exam, one which are numerical based and the other theoretical. In numericals, primarily weightage is given to accuracy. If the final answer is not correct, then at least 50% of the marks are deducted. Here, then, whether some marks are allotted or not will depend upon whether the candidate has written the solution systematically in steps or not and the step at which the calculation mistake has been made. Depending on the stage, marks can then be given accordingly. If, however, a conceptual mistake is made, then no marks will be granted.

In theoretical questions, step marking is definitely present as these involve writing proofs of the theorems or, in applications based on theorems, marks are dedicated to how and when the relevant theorems or results are used and quoted. Stating and proving lemmas and incorporating reasons for each step, as well as the flow of the entire solution, helps the examiner to allocate marks to the steps attempted.

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